The paper deals with generalized functional regression. The aim is to
estimate the influence of covariates on observations, drawn from an exponential
distribution. The link considered has a semiparametric expression: if we are
interested in a functional influence of some covariates, we authorize others to
be modeled linearly. We thus consider a generalized partially linear regression
model with unknown regression coefficients and an unknown nonparametric
function. We present a maximum penalized likelihood procedure to estimate the
components of the model introducing penalty based wavelet estimators.
Asymptotic rates of the estimates of both the parametric and the nonparametric
part of the model are given and quasi-minimax optimality is obtained under
usual conditions in literature. We establish in particular that the LASSO
penalty leads to an adaptive estimation with respect to the regularity of the
estimated function. An algorithm based on backfitting and Fisher-scoring is
also proposed for implementation. Simulations are used to illustrate the finite
sample behaviour, including a comparison with kernel and splines based methods